>>> Bob Sewell on Infinity 
 BS> Yes, I've seen the proof for this.  It shows that the cardinality
 BS> of the real numbers between any integer n and its neighbor integer n +
 BS> 1 (or n - 1) are uncountably infinite.  Understanding this proof on a
 BS> rational, logical level doesn't make it any easier for my mind to
 BS> comprehend an infinity larger than another on the level of common
 BS> sense.  Because I can never empirically perceive *any* infinity, I
 BS> can't empirically perceive one infinity larger than another.
Start with finite analogy.  Let all Si (S sub i, often notated S_i) be sets 
of n elements, and consider choosing an element from each Si.  If you have 
only S1 to choose from, you can choose in n different ways.  If you can 
choose from both S1 and S2, you can choose in n^2 different ways making n^2 
different ordered pairs S1 x S2.  You can choose one element from each of S1, 
S2, ..., Sk, in n^k different ways, creating n^k different order k-tuples.  
Using D for denumerable, when n = D, you are making an infinite number of 
choices, D each from a denumerable set Si for i = 1,2,3,..., in D^D different 
ways, creating D^D different D-tuples or series of integers.  An infinity of 
choices infinitely many times all at once.
As we've discussed D^D > D, showing the number of sequences of positive 
integers, n1,n2,n3,... is greater than the number of positive integers, 
1,2,3,....  For indeed, a sequence is infinitely more than a single integer.
 BS> It seems to be the same problem many people have with the idea of
 BS> God.  They cannot or will not believe in what they cannot empirically
 BS> perceive.
The same paradoxes about god are the same that have bother infinity.
Can god do something greater than anything an omnipotent being can do?
Can the set of all sets not belonging to itself, belong to itself?
Can there be an infinite ordinal greater than all infinite ordinals?
Can an infinity bigger than anything be bigger than infinity?
 BS> Sorry.  All this contemplation of infinities makes my brain hurt.
 BS> It's a small wonder Cantor went insane.
What?  You know his history?  Please tell.  I read his original thesis upon 
transfinite numbers when a kid.  Thanks to you, -) I'm borrowing a library 
copy of his original work.  Maybe it will have some insights to help you.  
I'll look.  It should make interesting discussion.  Maybe even philosophical.
 BS> And difficult to understand.
The difference between cardinals and ordinals is not hard.  What's hard is 
the plethora of ordinals. 
 BS> So, I guess you are right and I was mistaken.  But, like I said, I
 BS> really don't believe there is any number past A because the very
 BS> definition of A seems to me to include any and all numbers greater
 BS> than A.
A?  Aleph-nul, A0?  But it's defined as countable, in a 1-1 correspondence 
with the integers.  What's your definition of A?
 
 WE> #1 = 0.a1 a2 a3 a4 ...
 WE> #2 = 0.b1 b2 b3 b4 ...
 WE> Now I show there is a real number #x not in this list.
 WE> This is the diagonal argument.
 BS> Yes, this is the proof to which I referred above.  However, my
 BS> Discrete Math textbook calls the proof of the countability of the set
 BS> of positive rational numbers which I showed in the last post to you
 BS> Cantor's diagonal proof.
I surmise there's a difference between a diagonal argument and a diagonal 
proof.
 
 WE> Do you know about P(S) the power set of S which the set of all sets
 WE> included in the set S, {x | x included in S}?  A simple generalized
 WE> diagonal argument can be used to show the the cardinality of P(S) is
 WE> greater than the cardinality of S itself. So we can continue S, P(S),
 WE> P(P(S)), etc with a never ending series of ever larger infinities. 
 WE> But that's not the end, the whole series can be leap frogged to an
 WE> even greater number than all of the previous numbers.  There's no end
 WE> to this series of ever greater infinities.
The interesting thing about proving cardinality of P(S) > cardinality of S is 
it's similarity with the Russell paradox:  Can the set of all sets not 
belonging to itself, belong to itself?
 BS> I know.  It still doesn't make it easier on a common sense level.
 BS> And it makes my brain hurt to try to make sense of it on that level.
Infinity is paradoxical.  You did understand how A0 + A0 = A0, so you see you 
do understand some paradoxical things about infinity.  For example, as Hindu 
scriptures say, 'From fullness take fullness and fullness remains.'  That was 
centuries before Cantor showed that taking odd integers away from all 
integers leaves the even integers leaving as much as what you started with.  
So you see, Cantor is a late comer to this realm.  
The infinite is beyond comprehension is what Christian mystics say.  What 
mathematicians say is there are some infinities so big that they are 
inexpressible by any formulas.  They are called inaccessible cardinals.  That 
is a profound and philosophically insightful discussion for another post.
Yes, the reaching of limits again and again is mind boggling.  First you 
create a ever increasing series of infinities.  You take that series to the 
limit, creating an infinite number greater than all of the numbers in the 
original series.
Then you start again with the new greater number g, creating 2^g, 2^2^g, ... 
etc in another ever increasing series of infinities and again take the limit 
creating yet a even greater greater infinite number G.  That's the second 
time you took the limit.
But wait!  You've done nothing.  Repeat using G to get G1, G1 to get G2, 
taking limits for the third and forth times.  But don't stop.  Do this an 
infinite number of times, G1, G2, ....
Now take this process (of taking limits to the limits) to the limit.
Get the idea?  Continue infinitely. limits of limits of limits, limits of 
limits of limits of limits, limits of... ***. (*** means ... repeated ... 
times).  Is your mind boggled by now?  It should be otherwise it wouldn't be 
infinite, now would it? -)
But I'm describing infinity, and what I've described so far is just a 
beginning, else it wouldn't be infinity, now would it. -)  How big can 
infinity get?  Take an infinite number N and create larger infinities N^N, 
N^N^N, ....  N times, and you've done next to nothing but express the 
expressible. -)  Yet the mystics admonish infinity is inexplicable.
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